PSI - Issue 38

Arvid Trapp et al. / Procedia Structural Integrity 38 (2022) 260–270 A. Trapp / Structural Integrity Procedia 00 (2021) 1–11

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(a) samples

(b) fit

Fig. 7: Three-dimensional sample space for { ν ∗ 0 , ν ∗ p , k }

3.5. Deriving an error function When considering a simultaneous representation for ν ∗ 0 and ν ∗ p it becomes apparent that both contribute to the description of the CPD (Fig. 7a). The three-dimensional scatter plot shows curved planes with only few scatter for each Miner exponent k . This motivates to derive an approximating function for the errors due to insu ffi cient sampling. To derive the error function the following specifications are considered. It must be based on the intro duced parameters ν ∗ 0 and ν ∗ p (Eq. 13) as they prove to be good descriptors and foremost include the sampling property in the form of the Nyquist frequency f Ny . To include a wide band of these parameters, realistic and synthetic PSDs were combined for a large data set, on whose basis the error function is derived. Hereby, the synthetic PSDs o ff er a wide range of parameters, while the realistic PSDs include realistic shapes. As the prior investigations have shown, the Miner exponent has considerable e ff ect on the results and is therefore included as another parameter. To summarize, the error function includes { ν ∗ 0 , ν ∗ p , k } and is defined for the CPD = 1 − ε , whereby ε is the error due to insu ffi cient sampling. Finally, the function is designed so that the CPD → 1 resp. ε → 0, when ν ∗ 0 , ν ∗ p → 0. The herein presented error functions (Eqs. 14, 15) were derived on the basis of 300 synthetic and 7200 realistic PSDs. For calculating the CPD a zero-padded time series with a zero-padding factor of ζ = 20 provides the reference. Approaching the curve-fitting problem, a first error formula was accomplished by neural net regression. This provided very good results with only few computational times. However, the error formula is not well interpretable. Therefore, we investigated a set of formulas with di ff erent configurations of the central parameters. Hereby some weighting factors were left as optimization variables, which were calculated by a nonlinear curve-fitting algorithm. Each promising solution of an error formula was judged by the number and the complexity of terms and by calculating the mean squared error (MSE). It became apparent that the majority of the sample space could be well approximated by a simple formula, CPD = 1 − ˆ ε = 1 − 3 ν ∗ 0 ν ∗ p − ( ν ∗ 0 ν ∗ p ) 2 5 + k (14) that includes the product of ν ∗ 0 and ν ∗ p . However, for small, but also for large Miner exponents, there is a tendency to more pronounced deviations (Fig. 8). On the one hand, for small Miner exponents, samples with a low α 2 value (resulting from a small ν ∗ 0 value in comparison to ν ∗ p ) lead to larger deviations. On the other hand, for larger Miner exponents, the deviations between samples and fit increase with ν ∗ p . Therefore, an extended formula is proposed which includes two additional terms that improve the CPD fit. CPD ext = 1 − ˆ ε = 1 − 3 ν ∗ 0 ν ∗ p − ( ν ∗ 0 ν ∗ p ) 2 5 + k − 1 40 (1 − α 2 ) ν ∗ 2 p k − 0 . 9 − k ν ∗ p 1000 (15) Finally, the error formulas (14) and (15) are evaluated. To provide a basic assessment of the approximation formulas, the residuals (root of MSE) between the sample set and the error formulas for each Miner exponent